Copula models with regression margins have been used widely; for examples, see [1]}, [2]}, [3]} and
[4]}.
However, another usage of a copula with regression data is to capture the dependence between multiple observations
on a single dependent variable \(Y\) , conditional on the covariate values.
This defines a copula process [5]} on the covariate
space, which [6]} call a “regression copula”. When combined with a flexible marginal distribution for \(Y\) , it
specifies a new distributional regression model. This is where
the covariates affect the entire distribution of \(Y\) .
[7]} and [6]} consider a regression copula
that is the implicit copula
of the joint distribution of observations in an auxiliary regression model.
They are inherently high dimensional, yet can be estimated in reasonable time
using Bayesian methods.
The idea is outlined in this section for continuous \(Y\) ,
and greater detail can be found in these papers.
